This article presents a highly accurate and efficient method—a discrete version of the preintegration method—for numerically solving the Cauchy problem for a singularly perturbed third-order equation. The method is based on expanding the highest derivative of the differential equation into a finite series in Chebyshev polynomials of the first kind with unknown expansion coefficients. All lower derivatives and an approximate solution to the differential equation are determined by preintegrating the series for the highest derivative using a discrete integration formula that reduces the order of the highest derivative. This yields the fundamental algebraic equations of the proposed method. By adding additional equations derived from three initial conditions to these fundamental equations, a system of linear algebraic equations is obtained for determining the unknown expansion coefficients for the proposed solution. The number of equations and the number of unknowns in the resulting algebraic system coincide. This system is solved by a standard method; in this paper, it is solved by the Gaussian method. The calculation results show that for arbitrary values of the small problem parameter, a slight increase in the number of Chebyshev polynomials leads to a geometrically progressive reduction in absolute errors. Thus, the proposed discrete version of the pre-integration method is not only computationally efficient but also highly accurate and sufficiently versatile for solving a wide range of problems involving singularly perturbed equations.
| Mualliflar | Shakaeva, E.E., Шакаева, Э.Э. |
|---|---|
| Jurnal | Ҳисоблаш ва амалий математика муаммолари |
| Nashriyot | Тошкент ахборот технологиялари университети қошидаги, Ахборот-коммуникация технологиялари илмий-инновацион, маркази |
| ISSN | 2181-8460, 2181-046X |
| Nashr sanasi | 2026-05-02 |
| Son | 2 |
| Betlar | 109-121 |
| Til | Ingliz |
| Ilmiy soha | Fizika-matematika |
| DOI | 10.71310/pcam.2_72.2026.07 |
| Indeksga qo‘shilgan | 2026-08-27 |
DOI: 10.71310/pcam.2_72.2026.07 · Maqolaning asl sahifasi
Chebyshev polynomials, preliminary integration method, discrete version, Cauchy problem, singularly perturbed equations, algebraic system, полиномы Чебышева, метод предварительного интегрирования, дискретный вариант, задача Коши, сингулярно возмущенные уравнения, алгебраическая система
| APA | Shakaeva, E.E., & Шакаева, Э.Э. (2026). Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation. Ҳисоблаш ва амалий математика муаммолари, (2), 109-121. https://doi.org/10.71310/pcam.2_72.2026.07 |
|---|---|
| GOST (OAK) | Shakaeva, E.E., Шакаева, Э.Э. Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation // Ҳисоблаш ва амалий математика муаммолари. – 2026. – № 2. – P. 109-121. – DOI: 10.71310/pcam.2_72.2026.07. |
| MLA | Shakaeva, E.E., and Шакаева, Э.Э. “Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation.” Ҳисоблаш ва амалий математика муаммолари, (2), 109-121, 2026. https://doi.org/10.71310/pcam.2_72.2026.07 |
| Chicago | Shakaeva, E.E., and Шакаева, Э.Э. “Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation.” Ҳисоблаш ва амалий математика муаммолари, (2), 109-121 (2026). https://doi.org/10.71310/pcam.2_72.2026.07 |
| Harvard | Shakaeva, E.E., and Шакаева, Э.Э (2026) ‘Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation’, Ҳисоблаш ва амалий математика муаммолари, (2), 109-121. https://doi.org/10.71310/pcam.2_72.2026.07 |
| BibTeX | @article{shakaevaee2026,
title = {Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation},
author = {Shakaeva, E.E. and Шакаева, Э.Э.},
journal = {Ҳисоблаш ва амалий математика муаммолари},
year = {2026},
number = {2},
pages = {109-121},
doi = {10.71310/pcam.2_72.2026.07},
url = {https://doi.org/10.71310/pcam.2_72.2026.07}
} |
Boshqa maqola uchun iqtibos tayyorlash
This paper examines a boundary value problem for a degenerate elliptic equation in a planar domain bounded by a line segment and an analytic curve. The primary objective is to construct an explicit analytical solution…
This research develops a mathematical formulation and numerical solution for the spatial-temporal distribution of groundwater head in heterogeneous porous media. Based on Darcy’s and mass conservation laws, a 3D…
Recent years have witnessed significant climatic changes and increasing environmental pressure globally, including in Uzbekistan, necessitating an objective regional assessment. This research develops an approach for…
This research presents a multidimensional mathematical model and a robust secondorder numerical algorithm for coupled heat and moisture transfer with pressure-driven gas flow. Utilizing fractional Caputo derivatives (0…
This article presents a mathematical model and algorithm for numerically solving a problem for studying the hydrodynamic process of in-situ leaching in a heterogeneous porous medium, taking into account mass transfer…
This paper considers a two-dimensional linear hyperbolic system with dynamic boundary conditions and proposes a difference scheme for its numerical solution. An explicit–implicit directional splitting method is…
Fourier integrals of functions appear primarily in the theory of special functions and Fourier analysis, but also in other applied and computational sciences and engineering, such as theoretical physics, acoustic…
In this paper, an optimal interpolation formula with derivative constructed in the Sobolev space is considered. The formula interpolates an unknown function using its values and derivatives first three order at equally…
The article considers hybrid CPU-GPU parallel implementations of numerical solution algorithms for the hereditary model equation of radon volume activity. The test example is a direct Cauchy problem for a nonlinear…
A three-dimensional mathematical model is proposed for the inflow of a turbid (sediment-laden) river flow into a density-stratified reservoir, accounting for phase separation and the breakdown of the layered structure…
Ҳисоблаш ва амалий математика муаммолари — barcha maqolalar · 2026-yil arxivi
The article presents a numerical modeling approach for solving a non-homogeneous singularly perturbed fourth-order equation using by a discrete variant of the pre integration method. In the proposed method, the basis…
This paper presents a highly accurate and efficient spectral-grid method for the numerical solution of a boundary value problem for a fourth-order ordinary differential equation with a small parameter at the highest…
This article presents the results of 2018-2023, such as biological diversity, inter-landscape distribution characteristics, density,and number of straggly insects identified from natural and anthropogenic areas of…
This article observational data on the species composition, distribution, population, density, and seasonal damage caused causedby herbivorus bugs. Observationes were carried out in April and September 2021. When…
In the article, the research fields of some species belonging to Paeonia, their significance in human life, adaptation features tovarious ecological conditions are highlighted, and the ecologist groups and other…
This article studies the number of species, population density of terrestrial mollusks of the Topalangdarya basin, located in theSurkhandarya region of Uzbekistan, in various altitudinal regions and biotopes and…
This article sheds light on changes in biomass formation and antimicrobial activity against test pathogenic strains under theinfluence of epigenetic modifiers and elicitors of the endophytic fungus Penicillium…
Shahimardonsoy covered issues such as the soil in the river basin and measures to increase its condition, salinity and fertility, theflora and fauna present in the basin landscape and their conservation.